UNIVERSITY  OF  CALIFORNIA  PUBLICATIONS 

IN 

AGRICULTURAL    SCIENCES 

Vol.  4,  No.  9,  pp.  233-245,  3  figures  in  text  December  20,   1921 


THE  ALINEMENT  CHART  METHOD  OF 
PREPARING  TREE  VOLUME  TABLES1 


BY 

DONALD  BRUCE 


The  chief  use  of  the  alinement  chart2  is  to  express  with  the  simplest 
possible  system  of  lines  a  law  the  equation  of  which  is  known.  The 
underlying  principle  is  so  flexible  that  almost  any  formula  can  be 
expressed  thereby,  although  the  most  striking  advantages  over  a  system 
of  rectangular  coordinates  do  not  appear  unless  three  or  more  variables 
are  involved.  The  axes  may  be  parallel  or  converging,  straight  or 
curved,  and  graduated  either  uniformly  or  with  intervals  which  vary 
in  accordance  with  some  given  law,  the  form  of  the  graph  depending 
on  the  form  of  the  equation.  It  follows  from  this  very  flexibility  that 
such  charts  are,  in  general,  unsuited  for  use  with  empirical  data.  The 
following  pages,  however,  describe  an  exception  to  this  general  rule  in 
which  one  type  of  alinement  chart  may  be  advantageously  used  in  the 
preparation  of  tree  volume  tables,  although  the  form  of  the  equation 
of  such  a  table  is  yet  unknown. 

The  most  suitable  type  of  chart  can  be  determined  by  working 
out  an  approximate  algebraic  expression  for  the  volume  of  a  tree  in 
terms  of  its  diameter  and  height.  This  expression  is  complicated  by 
the  fact  that,  for  almost  all  American  tables,  volumes  must  be  com- 
puted in  board  feet  as  scaled  by  some  log  rule,  instead  of  in  cubic  feet. 
The  starting  point  must  therefore  be  the  equation  of  the  volume  of  a 


1  Acknowledgment  is  made  to  Professor  Frank  Irwin,  of  the  Department  of 
Mathematics  of  this  University,  for  assistance  in  connection  with  the  analytic 
features  of  this  study. 

2  A  complete  discussion  of  the  theory  of  alinement  charts  may  be  found  in 
such  works  as:  J.  Lipka,  Graphical  and  Mechanical  Computation;  J.  B.  Peddle, 
The  Construction  of  Graphic  Charts;  and  M.  d'Ocagne,  Traite  de  Nomographic. 
For  a  discussion  of  the  application  of  certain  simple  types  to  some  formulae  of 
forest  mensuration  see  D.  Bruce,  "Alinement  Charts  in  Forest  Mensuration," 
Journal  of  Forestry,  XVII,  7,  773  (November,  1919). 


\ 


234  University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 

log  in  board  feet  first  formulated  by  Professor  Daniels,3  i.e.,  v  =  ad'2 
+  bd  +  c  (I).  For  simplicity,  let  us  apply  this  formula  to  a  tree  of 
uniform  taper  up  to  its  merchantable  top  limit,  that  is,  one  which  is  a 
frustum  of  a  cone. 

Let  V  =  volume  of  a  tree  in  feet  b.  m. 

Let  v  =  volume  of  a  log  in  feet  b.  m. 

Let  D  =  d.  i.  b.  stump  (assumed  equivalent  to  d.  b.  h.). 

Let  d  =  top  diameter  of  a  log. 

Let  H  =  height  in  logs  of  tree. 

Let  t  =  top  d.  i.  b.  of  tree. 

It  is  evident  that  the  taper  of  the  tree  =  D  —  t,  and  that  the  taper 

per  log  =  — z- —    Therefore  the  top   diameters  of  the  several  logs 

.  D—t 

of  the  trees  are  the  terms  of    the   following  series :    t,  t    +   — fj~} 

t  _j_  2(^~0  f  t  +  3(D—t)}  to  jj  terms,  and  the  volumes  of  the 

H  H 

same  are  the  terms  of  the  following  series,  each  top  diameter  being 

successively  substituted  in  I : 
at2+bt+c; 

at'  +j±  (D  -  t)  +  jp(D  -  ty  +  bt  +  |  (D  -  t)  +  c; 
at'  +  *£(D-t)+  J  (D  -ty  +  U  +  2^(D-t)+  c; 
at2  +  to*  {D  _  t)  +  g  {D  _  ty  +  bt  +  g  {D  _  t)  +  c; 

at2  +  8J  {D  _  t)  +  go  (Z)  _0,  +  w  +  «(jD_0  +  c. 

.  .  .  .  to  H  terms. 

V  =  the  sum  of  this  series  to  H  terms.  This  may  be  obtained  by  the 
differential  method  in  which  a  new  series  of  first  differences  is  derived 
by  subtracting  each  term  from  that  which  follows  it,  and  this  process 
is  repeated,  successively  obtaining  a  series  of  second  differences,  third 
differences,  etc.,  until  all  terms  become  zero  and  the  series  vanishes. 


-Sec  A.  L.  Daniels,  Measurement  of  Sawlogs,  Vermont  A.gr.  Exp.  Sta.,  Bulletin 
L02,  L903. 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      235 

m,              ,,                ,             .    11  (n  —  1)    ,     .       (n — 1)   (n — 2)    ,    , 
Ine  sum  then  equals  na    -\ ^r dx  -\-n ~ -  a\  + 

LA  LA 

....  where  n  =  number  of  terms,  a  =  the  first  term  of  the  original 
series,  d1  =  the  first  term  of  series  of  first  differences,  d2  =  the  first 
term  of  series  of  second  differences,  etc. 

Series  of  first  differences  is : 

^ .  (D  -  t)  +  jp  (D  -  ty  +A  (Z)  -  0; 

2f(D-t)+^(D-ty+±(D-t); 
|-<  (D  -  t)  +g(Z>  -  ty  +jj  (D  -  t); 

~  (D  -  I)  +  J  (Z)  -  ty  +  ~  (D  -  t);  etc. 


Series  of  second  differences  is : 

2a  (D  -  t)\    2a  (D  -  t)\    2a  (D  -  t)2 
H2         ;  H2         ;  H2 

Series  of  the  third  differences  is : 
o;                o;                o;  etc. 
And  V  =  sum  of  this  series   =  H  (at2  +  bt  +  c)   +  — — t> 

^(D  -  t)  +%(D  -  ty-  +  ±(d  -  t)}+H  (H  ~  ^H  -V 

\%(D-ty. 


;  etc. 


Expanding  and  rearranging  in  terms  of  D,  this  becomes : 

.    f  (2afi  +  Zbt  +  &c)H   .    {Zaf-  +  3bt)        at2  . 
+  6  +  6~~    "  +  6/7  ' 


\ 


236  University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 

Rearranging  in  terms  of  H,  this  may  also  be  written : 

V=^{  2aD2  +  (2at  +  35)  D  +  2a/2  +  Sbt  +  6c  }  —  Y2  j  aD2  + 


ID 


—  t(at+b)   \  H (D2  —  2tD  +  t2)  (III) 

j        6# 

Let  us  now  apply  this  general  formula  to  a  specific  case,  for  example, 
that  of  trees  scaled  by  the  Scribner  log  rule  to  a  six-inch  top  cutting 
limit.  A  close  approximation  formula  for  this  log  rule  (for  16-foot 
lengths)  is: 

V  =  .765d2  —  .55d  — 21. 

We  therefore  may  assume 

a  =.765 
b=—  .55 

c  =  —  21 

Substituting  these  values  in  III,  we  have : 

V  =  H  (.255Z>2  +  1.2550  —  13.47)  —  (.3825D2  —  .275/)  — 

1 
12.12)  +  —  (.1275Z)2  —  1.532)  +  4.59) .  (IV) 

H 

Typical  equations  for  the  height  class  curves  of  a  volume  table  in 
graphic  form  can  now  be  found  by  substituting  in  IV  given  values 
of  H;  for  example : 

For  H  =  2,  F  =  .19125Z)2  +  2.02Z)  —  12.52  (V) 

H=6,  V  =  1.16875/)2-}-7.55/)  —  67.93  (VI) 

H  =  10,  V=  2.18025/)2  +  12.672/)  —  122.121  (VII) 

Similarly,  typical  diameter  class  curves  are : 

9  04 
For  D  =  10,  V  =  24.58/7  -  23.38  +  n^  (VIII) 

ti 

D  =  20,  V  =  113.63//  -  135.4  +  ^°  (IX) 

73  4-4- 
D  =  30,  V  =  253.68//  -  323.88  +  -^P  (X) 

94  fi  S4 
/)  =  50,  7  =  686.78//  -  930.38  +  =^£^  (XI) 

ti 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      237 

It  will  readibly  be  seen  that  V,  VI,  and  VII  are  equations  of  parabolas, 
while  VIII  and  IX  and  X  and  XI  are  hyperbolas.  These  deductions 
agree  so  well  with  the  actual  results  obtained  in  volume  tables  con- 
structed by  the  conventional  method  on  a  similar  basis  that  it  seems 
probable  that  the  general  form  of  the  equation  should  apply  at  least 
approximately  to  actual  trees  as  well  as  to  the  cone  frusta  on  which 
it  is  based.  Furthermore,  it  has  been  tentatively  established,  and  with- 
out any  conflicting  evidence  coming  to  light,  that,  in  the  case  at  least 
where  a  fixed  top  cutting  limit  is  used,  frustum  form  factors  are  func- 
tions of  diameter  and  not  of  height.  If  this  is  true,  such  equations  as 
VIII  can  be  corrected  to  apply  accurately  to  any  given  species  by 
multiplying  into  them  the  proper  form  factors,  which  would  merely 
change  the  values  of  the  constants  without  affecting  the  form.  Finally, 
the  ease  with  which  the  alinement  chart  devised  to  apply  to  cone 
frusta  works  out  for  actual  trees  is  the  best  proof  of  the  adequacy  of 
the  equation. 

Next,  it  is  necessary  to  determine  this  alinement  form.  Unfortu- 
nately a  difficulty  at  once  presents  itself.  The  equation  appears  to  be 
one  of  those  rare  instances  which  cannot  be  thus  expressed.4  It  has 
been  found  by  experiment,  however,  that  if  two  parallel  axes  be  as- 
signed to  V  and  H,  the  former  graduated  uniformly  upward  and  the 
latter  uniformly  downward,  all  lines  expressing  a  single  value  of  D 
(taken  from  a  table  of  values  of  volumes  of  cone  frusta  in  board  feet 
or  calculated  by  the  above  formulae)  will  intersect  nearly  (but  not 
quite)  in  a  common  point,  and  that  these  common  points  for  a  series 
of  values  of  D  lie  almost  (but  not  quite)  in  a  straight  line,  which  if 
produced  will  pass  through  the  zero  point  on  the  V  axis.  Figure  1 
illustrates  this  fact,  although  a  larger  scale  is  needed  to  bring  out  the 
failure  of  the  lines  to  intersect  perfectly. 

The  reason  for  this  becomes  evident  upon  analysis.  Let  the  lower 
left-hand  corner  of  figure  1  serve  also  as  the  origin  of  a  system  of 
rectangular  coordinates  with  one  unit  equaling  ten  of  the  small  squares. 
Also  let  b  equal  the  width  of  the  paper.  Then  any  straight  line 
used   in   solving   values   by   the   alinement   chart   can   be    expressed 


4  Only  those  equations  can  be  expressed  by  an  alinement  chart  that  can  be 
put  in  the  determinant  form 

A       Or       h 

f2         g2         Ju     =  0 
U         9z         \ 

where  fi,  gi,  and  h-t  are  functions  of  x,  (i  =  1,  2,  3). 


238  University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 


f     v) 

as  connecting  the  points  (0,  22  —  2H)  and-<  b,  —^  for  by  the  equation 


100 


I   V  1    X 

Y=  \~q  -  22  +  2H      y  +  22  -  2#. 


r 


(XI) 


10 


.fcf 

'S3 


Figure  1 

Alinement  Form  giving  approximately  correct  results  for  volumes 
of  cone  frusta  in  feet  b.m. 


O   c3 
> 


Now  from  equation  IV,  V  =  AII  -\-  B  -\ (where  A,  B,  C  are  func- 

II 
lions  of  D)  and  the  equations  of  two  such  lines  corresponding  to  any 
values  oi*  //,  such  as  //,  and  II 2,  and  having  a  common  value  of  D,  will 
then  be  (from  equation  XI)  : 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      239 


Y  = 


AH1  +  B  + 


J7 


and 


100 


AH2  +  B  + 


(I 


100 


-  22  +  2H2 


X 


-  22  +  2#!  I    6 


+  22  -  2#i 


X 


+  22  -  2#2 


The  point  of  intersection  of  these  two  lines  can  now  be  found  by 
solving  these  two  equations  simultaneously,  and  when  this  is  done  the 
following  values  of  X  and  Y  are  obtained : 


X 


2006 


A  - 


C 


Hi  Hi 


+  200 


C 


-  =^=.  !>   +  25  +  ^  +  ^ 
ill  £li  tli        tl<i 


Y  = 


A  - 


C 


Hi  Hi 


+  200 


(XIII) 


(XIV) 


It  will  be  seen  that,  for  such  a  range  of  values  for  D  and  H  as  are 
actually  encountered,  these  two  equations  approximate  quite  closely  to 


X  = 


2006 


A  +  200 


22A  +  2ff 

A+  200 


(XV) 


(XVI) 


Since  both  X  and  Y  in  this  last  pair  of  equations  are  independent  of 
H,  this  approximation  explains  the  approximate  common  intersection 
of  all  lines  having  a  given  value  of  D.  They  can,  moreover,  be  com- 
bined to  give  an  equation  of  the  curve  upon  which  all  these  common 
intersections  fall,  but  the  result  is  in  a  form  too  complicated  to  be  of 
much  value,  although  it  can  be  readily  identified  as  an  equation  of  a 
conic  section  (obviously  a  very  straight  portion  of  a  hyperbola).  The 
curve  can  be  plotted  more  simply  by  means  of  equations  XV  and  XVI, 
and  will  be  found  to  be  very  nearly  a  straight  line  passing  through 
the  zero  point  on  the  V  axis. 


240  University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 

It  seems  as  if  a  regraduation  of  the  H  axis  might  be  made  to  result 
in  perfect  instead  of  approximate  intersections.  It  will  be  found  that 
this  can  be  readily  accomplished  for  any  given  value  of  D  on  the  as- 
sumption that  the  H  graduating  distance  from  a  fixed  point  =  KH 

M 

-\-  L  -| where  K,  L,  and  M  are  constants.    But  unfortunately  these 

H 
three  constants  prove  to  be  themselves  functions  of  D.    In  other  words, 

different  sets  of  graduations  would  have  to  be  used  for  each  value  of 
the  diameter,  which  is  obviously  impracticable. 

Each  value  of  H,  however,  is  in  practice  associated  with  a  rather 
narrow  range  of  D  values.  It  is  therefore  possible  to  modify  the 
positions  of  the  H  graduations  empirically  so  as  to  result  in  a  decided 
improvement  in  the  intersections,  and  at  the  same  time  a  slight  re- 
adjustment of  the  D  axis  can  be  made  with  advantage.  The  best 
results  appear  to  be  obtained  by  the  following  plan : 

1.  Graduate  the  V  axis  as  already  described. 

2.  Graduate  the  H  axis  as  already  described  but  omit  all  values 
under  that  of  5  logs. 

3.  Select  a  few  definite  values  of  D  well  distributed  over  the  de- 
sired range,  and  for  each  calculate  the  volumes  for  three  or  more  values 
of  H  (H  =  5  or  over).  If  a  table  of  cone  frusta  is  available,  these 
volumes  may,  of  course,  be  taken  therefrom. 

4.  Draw  straight  lines  from  each  value  of  H  to  the  corresponding 
value  of  V,  and  select  points  which  appear  to  be  the  averages  of  the 
intersections  of  lines  relating  to  each  common  value  of  D. 

5.  Draw  a  curve  through  the  points  thus  selected.  For  most  work 
a  sufficiently  close  approximation  will  be  found  to  be  a  straight  line 
passing  through  the  V  origin. 

6.  Enter  on  this  curve  or  straight  line  the  D  graduations  indicated 
by  the  straight  lines  of  step  4. 

7.  Obtain  two  or  three  values  for  V  corresponding  to  the  lower 
values  of  H  (under  5  logs)  and  to  the  values  of  D  already  graduated. 
By  drawing  the  appropriate  straight  lines  their  intersections  with  the 
II  axis  will  indicate  the  best  average  positions  of  the  smaller  H  gradu- 
ations.   These  should  then  be  entered  on  that  axis. 

8.  Complete  the  graduation  of  the  D  axis  by  intersection,  using 
for  each  value  of  D  appropriate  values  of  .IT. 

The  following  table,  worked  out  by  the  above  process,  may  also  be 
used  directly  to  save  time  and  labor. 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      241 


TABLE  I 

Graduation  of  H  Axis 

logs 
with  th< 

Distance  from  fixed  point  on  axis 

A 

Height  in 
tersection 

r 

Values  to  be  used 

where  the  heights 

are  in  logs  and 

tenths  of  logs 

3 

Values  to  be  used 

where  the  heights 

are  in  feet 

diagonal 

axis 

1.26 

20.16 

2 

•2.08 

33.28 

3 

3.03 

48.48 

4 

4.01 

64.16 

5 

5 

80 

6 

6 

96 

7 

7 

112 

8 

8 

128 

9 

9 

144 

10 

10 

160 

11 

11 

176 

In  the  application  of  this  theory  to  the  making  of  a  volume  table 
the  successive  steps  may  be  as  follows : 

1.  Prepare  an  incomplete  alinement  graph  (fig.  2)  such  as  has 
been  illustrated  in  Figure  1,  but  with  the  H  axis  graduated  in  accord- 
ance with  the  table  just  given. 

2.  Draw  a  diagonal  straight  line  representing  the  D  axis  between 
the  point  representing  its  intersection  with  the  H  axis  and  the  zero 
point  on  the  V  axis. 

3.  Graduate  the  D  axis  as  follows:  Each  tree  measurement  (a 
sufficient  number  of  which  are  supposed  to  be  at  hand)  is  used  to  draw 
a  straight  line  between  the  point  on  the  H  axis  corresponding  to  the 
height  of  the  tree  and  the  point  on  the  V  axis  corresponding  to  the 
volume  of  the  tree.  The  intersection  of  this  with  the  D  axis  is  an 
indication  of  the  position  of  the  D  graduation  corresponding  to  its 
diameter.  The  indications  of  a  number  of  trees  will  naturally  be 
more  or  less  conflicting  and  the  results  must  therefore  be  evened  off 
by  a  graduating  curve  such  as  indicated  in  figure  3.  In  this  curve 
the  distance  of  each  intersection  above  the  base  of  the  graph  of  figure 
2  is  plotted  over  its  corresponding  diameter.  When  all  the  points 
are  thus  plotted  a  smooth  curve  is  drawn  through  them,  and  from  this 
curve  the  D  axis  is  finally  graduated. 


242 


University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 


4      £ 


10 


'53 


7      -— 


Figure  2 
An  alinement  volume  table. 


O   °3 


Table  II  gives  the  basic  tree  data  used  in  this  illustrative  instance. 
The  broken  line  in  figure  2  shows  how  the  first  values  of  this  table 
are  plotted.  The  left-hand  point  on  figure  3  is  plotted  from  the  re- 
sulting intersection  with  the  D  axis. 

It  is  immaterial  whether  each  tree  is  thus  used  to  determine  a  point 
on  the  graduating  curve  or  whether  the  average  volume,  height,  and 
diameter  of  the  height-diameter  classes  are  used.  In  the  latter  case, 
each  point  should,  of  course,  be  weighted  in  accordance  with  the  num- 
ber of  trees  which  are  thus  averaged  together.  Figure  3  was  drawn 
in  accordance  with  the  latter  method. 

4.  The  volume  table  can  now  be  read  from  the  completed  alinement 
chart,  the  result  being  given  in  Table  III. 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      243 

TABLE  II 

Basic  Tree  Data,  White  Fir,  Stanislaus  National  Forest 

(Measurements  taken  by  U.  S.  Forest  Service) 


o.  of  trees 
in  class 

Average 
D.  B.  H.  inches 

Average 

merchantable 

height  in  16-ft. 

logs 

Average 
Vol.  ft.  B.  M. 

7 

18.1 

4.0 

280 

3 

19.5 

3.9 

330 

8 

19.9 

4.0 

340 

5 

19.8 

5.0 

500 

6 

21.9 

4.4 

540 

7 

22.2 

5.1 

640 

6 

21.8 

6.0 

670 

1 

22.8 

6.5 

810 

2 

23.9 

4.1 

510 

1 

23.4 

4.2 

490 

9 

23.7 

5.2 

640 

6 

24.1 

6.2 

970 

2 

24.6 

7.3 

1230 

4 

25.5 

5.0 

660 

9 

25.9 

5.6 

950 

12 

26.3 

6.0 

970 

9 

26.2 

7.0 

1270 

1 

25.8 

7.6 

1510 

4 

27.7 

5.6 

1050 

10 

28.3 

6.3 

1250 

7 

28.1 

7.5 

1450 

1 

27.6 

8.3 

1760 

1 

29.2 

4.6 

1090 

7 

29.9 

6.0 

1300 

7 

29.8 

6.4 

1280 

12 

29.7 

7.3 

1640 

3 

30.2 

7.9 

1720 

16  31.6  6.6  1600 

TABLE  III 

Volume  Table  Eead  From  Figure  2 

Height  in  16-ft.  logs 


.  B.  H. 

r 
4 

5 

6 

7 

8 

18 

280 

380 

20 

350 

480 

22 

430 

590 

740 

900 

24 

510 

690 

880 

1060 

26 

790 

1000 

1220 

1430 

28 

850 

1130 

1370 

1600 

30 

1010 

1280 

1550 

1820 

32 

1140 

1450 

1750 

2060 

244 


University  of  California  Publications  in  Agricultural  Sciences       [Vol.  4 


In  most  cases  it  will  be  found  that  the  graduating  curve  of  figure 
3  can  be  entered  on  the  same  sheet  as  the  alinement  chart  without 
confusion  and  with  a  saving  in  time  and  convenience.  In  step  3, 
moreover,  it  is  not  necessary  actually  to  draw  the  various  straight  lines, 
which  are  apt  to  become  confusing  if  many  tree  measurements  are 
available.  Instead,  a  straight  edge  may  be  laid  across  the  proper  values 
and  its  intersections' with  the  intermediate  axis  noted. 


1300 
1200 

1100 

1000 

900 

"1  800 
I  700 
5   600 


■■■     "  :t'-il  •■■:!"' Ij    !■--:'  •;!    i--|:"l    I  ■■■-,■  ■    I  ■;  "I  :::  i    J    V- 


18     19     20    21     22    23     24    25     26    27     28     29     30     31     32 
D.B.H.  inches 

Figure  3 

The  graduating  curve. 


There  are  several  advantages  in  this  method  of  preparing  a  volume 
table.  In  the  first  place,  the  curve  drawing  is  simplified.  In  place 
of  the  system  of  curves  which  have  to  be  harmonized  in  the  usual  plan 
only  a  single  graduating  curve  need  be  drawn,  and  since  this  is  based 
on  all  of  the  tree  measurements  available  it  is  much  better  denned 
and  more  easily  and  accurately  located.  As  compared  with  the  frus- 
tum form  factor  method,  which  also  uses  a  single  curve,  the  use  of  the 
alinement  method  saves  considerable  time,  since  the  calculation  of  the 
form  factors  is  avoided;  the  result  is,  however,  practically  identical, 
for  if  the  frustum  form  factors  of  such  a  table  as  Table  III  be  calcu- 
lated they  will  be  found  to  vary  with  diameter  but  not  materially  with 
height. 

Secondly,  exterpolations  are  easily  (perhaps  almost  too  easily) 
made,  especially  in  height,  and  with  far  more  certainty  than  is  possible 
by  the  normal  system  of  curve  extension. 

Lastly,  the  resulting  alinement  chart  can  be  read  with  great  accu- 
racy for  fractions  of  inches  in  diameter  and  fractions  of  logs  in  height. 


1921]       Bruce:  Alinement  Chart  Method  of  Preparing  Tree  Volume  Tables      245 

This  is  not  true  of  the  conventional  method,  where  graphic  interpo- 
lations between  the  harmonized  curves  are  both  slow  and  inaccurate 
and  where  arithmetical  interpolations  in  the  final  table  are  exceed- 
ingly laborious.  For  certain  problems  of  forest  mensuration  this 
advantage  is  highly  important,  although  it  is  of  little  weight  in 
connection  with  ordinary  timber  cruising.  The  method  appears  su- 
perior in  accuracy  to  the  ordinary  plan,  especially  where  the  amount 
of  data  available  is  limited.  Volume  tables  have  been  made  by  both 
methods  from  tree  data  for  three  species,  including  the  species  used  in 
illustrating  this  paper.    The  results  appear  as  follows: 

Aggregate  difference  between         Average   deviation  between 

all  trees  as  actually  scaled  individual  tree  volumes  as 

and  as  read  by  table  scaled  and  as  read  by  table 

. A .  . A . 

/■  ^  /■  ~s 

Basic  data  Conventional        Alinement  Conventional        Alinement 

145  trees,  western  larch 1.5%  0.2%  5.8%  5.1% 

166  trees,  western  white  pine 2.1%  0.3%  3.8%  3.9% 

166  trees,  white  fir 0.5%  0.2%  7.1%  6.3% 

It  will  be  observed  that  the  result  by  the  alinement  method  is  much 
superior  as  a  whole  for  each  species,  and  is  better,  on  the  average,  in 
detail  as  well. 


